Question

2x+1/x-2 + x-3/x+2= x^2+16/x^2-4

Answer 1

Ответ:

Пошаговое объяснение:

Answer 2

To solve the given equation:

2x + (1/(x - 2)) + ((x - 3)/(x + 2)) = ((x^2 + 16)/(x^2 - 4))

First, let's simplify the equation by finding a common denominator. The common denominator for the fractions is (x^2 - 4).

Multiply the first fraction by (x + 2)/(x + 2): ((2x)(x + 2) + (1/(x - 2))(x + 2) + (x - 3))/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Expanding the numerator of the first fraction: (2x^2 + 4x + (x + 2)/(x - 2)(x + 2) + (x - 3))/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Multiply the second fraction by (x - 2)/(x - 2): (2x^2 + 4x + (x + 2)(x - 2) + (x - 3))/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Simplifying the numerator of the second fraction: (2x^2 + 4x + (x^2 - 4) + (x - 3))/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Combining like terms: (2x^2 + 4x + x^2 - 4 + x - 3)/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Simplifying further: (3x^2 + 5x - 7)/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Since the denominators are the same, we can cancel them out:

3x^2 + 5x - 7 = x^2 + 16

Moving all terms to one side: 3x^2 + 5x - 7 - x^2 - 16 = 0

Combining like terms: 2x^2 + 5x - 23 = 0

At this point, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. However, the given equation does not factor easily, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 2, b = 5, and c = -23:

x = (-5 ± √(5^2 - 4(2)(-23))) / (2(2))

Simplifying inside the square root: x = (-5 ± √(25 + 184)) / 4 x = (-5 ± √(209)) / 4

Therefore, the solutions for the given equation are:

x = (-5 + √(209)) / 4 x = (-5 - √(209)) / 4